To be prepared, mentally and/or physically, for extreme opposition; typically used in reference to an aggressive or potentially violent situation.

The phrase originates with American hunters and woodsmen in regions frequented by the brown bear. Brown bears are the largest land-based predator on earth, and when expecting to deal with them the hunters would bring much more powerful rifles than they would if hunting other game.

When the police responded to the reports of a men in body armor with AK-47s robbing a bank, they came loaded for bear.

Carrying more equipment than necessary. Overloaded. It can be used in a variety of ways meaning a person is equipped more than necessary for his/her prey.

A woman dresses "to the nines" with high heels, lipstick, new hair-do, perfume, etc. for club-hopping. "She's loaded for bear."
A youth is going hunting for squirrel but takes his father's high-powered rifle and a shotgun. He's loaded for bear.